Grade 12Integral Calculus

∫cos(2 cot-1{ root(1-x/1+x)} = ????

Nikhitha H Jagadeesh

11 Years agoGrade 12

2 Answers

Jitender Singh

11 Years ago

Ans:
Hello Student,
Please find answer to your question below

$I = \int cos(2cot^{-1}\sqrt{\frac{1-x}{1+x}})dx$
$x = cos\theta$
$dx = -sin\theta d\theta$
$I = \int cos(2cot^{-1}\sqrt{\frac{1-cos\theta }{1+cos\theta }})(-sin\theta )d\theta$
$I = \int cos(2cot^{-1}(tan\frac{\theta }{2}))(-sin\theta )d\theta$
$I = \int cos(2cot^{-1}(cot(\frac{\pi }{2}-\frac{\theta }{2}))(-sin\theta )d\theta$
$I = \int cos(\pi -\theta )(-sin\theta )d\theta$
$I = \int -cos(\theta )(-sin\theta )d\theta$
$I = \frac{1}{2}\int 2cos(\theta )(sin\theta )d\theta$
$I = \frac{1}{2}\int (sin2\theta )d\theta$
$I = \frac{-cos2\theta }{4} + c$
$I = \frac{-cos(2cos^{-1}x) }{4} + c$